Posts Tagged ‘pentominoes’

On the Polyforms list, Erich Friedman posed a very interesting new pentomino tiling problem:

Tile a rectangle of minimal area with pentominoes so that for each pentomino there is exactly one stratum, or cluster of one or more copies of that pentomino that reaches from one side of the rectangle to the opposite side. Pentominoes in a stratum must form a single group, connected by edges, not just corners.

Michael Reid found this 3×30 solution:

It’s not hard to prove that it is minimal. A natural extension of the problem is to find minimal solutions for 4×n and 5×n rectangles. Michael Reid found the first 5×n solution, but I improved on it with this 5×32 solution:

The 4×n problem seems to be the hardest, and initially it was not clear that it would be possible. The X pentomino has only one possible stratum, which only can only be bordered by Y, I or N, and it is also difficult to find matches for a Z stratum. Additionally, only Y, L, and P can form straight line stratum boundaries usable for the top and bottom of the rectangle. (See wikipedia’s pentomino page if you don’t know the correspondence between letters and shapes.) I did eventually find this 4×50 solution:

This solution seems rather prolifigate with its pentominoes, but finding any solution at all was a bit of a surprise.

Update: Erich Friedman’s Math Magic for April 2010 further explored this subject.

So it should come as little surprise that I was intrigued by the cover of Puzzle Fun 16. Puzzle Fun was a ‘zine produced by Rodolfo Marcelo Kurchan in the ’90s covering a variety of polyomino problems. I missed out on subscribing to it myself, and the Puzzle Fun website languished for a decade after new issues stopped appearing.

Puzzle Fun 16 focused on pentomino packing problems. Packing problems differ from tiling problems in that empty space is allowed, and the goal is to minimize the amount of empty space required. Packing, usefully, makes some kinds of problems possible to solve that would not be solvable as tiling problems.

One such puzzle type is packing polyforms that are 2-colorable, (that is, one can use two colors to color every piece such that no piece touches another of the same color.) This is the puzzle type I saw on the cover of Puzzle Fun 16.

The problem itself was this: Place two sets of pentominoes in the smallest possible rectangle such that no pentomino touches another in the same set. [PF problem #549]

I should note that this problem implies strict coloring: pieces are not allowed to touch even at corners. I am more interested in non-strict coloring, which is generally the default in coloring problems, and I am interested in colorings of a single set of pentominoes. (Which all of the problems on my pentomino coloring page are.)

#1: Place a 2-colorable (non-strict coloring) set of pentominoes in the smallest possible rectangle. My best attempt has 65 squares:

Filling out the matrix of variations gives two more problems:

#2: As #1, but with a strict coloring.

#3: As in PF #549, but with a non-strict coloring.

(The following problem in PF 16 (#550) was a variation on #549 minimizing perimeter rather than area, but this is less interesting to me.)

I have had a lifelong interest in recreational mathematics, especially polyomino problems. I’ve produced some puzzles in laser-cut plastic, which I sell via this very site. I’ve also dabbled in writing interactive fiction.